Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4267 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,12)(13,21)(14,18)(15,16)(17,22)(19,23)(20,24)(26,31)(27,34)(29,37)(36,38);(1,2)(3,6)(4,7)(5,8)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,26,30,31)(27,28,34,32)(29,33,37,39)(35,36,40,38);(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,38,39,36);(1,3,5,4)(2,6,8,7)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,28)(26,27)(29,38)(30,32)(31,34)(33,40)(35,39)(36,37);(1,4,5,3)(2,7,8,6)(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,29,30,37)(26,33,31,39)(27,35,34,40)(28,36,32,38)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7944745);